Socio-Ecological Systems and Social Influence Network Modeling

Conceptual Model SNIP Model Details Model Operation Self-Organization and Learning

Social Influence Networks


Example Fire Network Nodes are actors/decision makers, edges are the connections between actors.

This site provides access to a web-based tool for modeling/exploring coupled socio-ecological systems (SES) through the use of social influence networks, defined as networks of individuals and/or institutions that take an input signal and process the signal through the network to generate a collective output signal, which is interpreted as a societal reponse to the landscape signal that potentially modifies the trajectory of that signal into the future. In this case, the two subsystems (social and ecological) of the SES are coupled, meaning they are part of a mutual feedback loop where each influences the behavior of the other.

The conceptual model being employed is described below.

The Network Conceptual Model

We model social influence networks (SIN) as a network of nodes and connecting edges. Network nodes represent individuals or institutions involved in influencing a decision. Edges represent pathways of influence propagation. Edge weights represent how effectively a given edge transmits a signal. The network is directional, meaning signals are propagated in one direction along a given edge. Each directional edge has a weight associated with it, that indicates the strength with which a signal is transmitted along that edge. The edge weight is computed dynamically based on an influence model described on the "SNIP Model Details" tab.

Component Definition
Signal An information flow that has specific qualities and intensity. We are concerned with signals in two contexts: landscape signals that represent information flows arising from landscape productions, and network signals that represent influence flows between actors in a network. Landscape signals can be any metric generated by Envision for which actor values are defined, and therefore can be static or dynamic. Network signals are determined based on Actor "reactivity" levels and influence flows in the network.
Network Actor an individual or other entity capable of processing information, located within a network with other actors.
Landscape Actor a network actor with control of one or more IDUs, and with the ability to select Envision policies for application in those IDUs (i.e. all actors are network actors, some are traditional Envision landscape actors
Actor Class A type associated with a given collection of actors. Envision currently uses Actor Classes to create individual landscape actors; the class is essentially a “template” used to populate landscape actors to specific IDUs. An Actor class defines individual actor values, which in turn affect actor policy selection. Note that in our recent conversations, we have assumed that network actors while not be class based, but that is perhaps a future conversation.
Cultural Trait a measure, on a continuous [-1,1] scale, of a cultural proclivity, e.g. Egalitarian, Individualistic, Communitarian, or Hierarchical, where -1 indicates a negative proclivity, and 1 indicates a positive proclivity, to the trait.
Cultural Trait Vector a set of Cultural Traits specific to a given Actor.
Social Network a directed topological graph of Network and Landscape Actors, possibly multilayer, through which signals are transmitted.
Edges Edges in the graph represent connections between Network Actors through which information flows in the form of signals. Edges may be uni- or bi-directional; indicating the direction of the flow of information between sender and receiver.
Nodes Nodes in the graph represent individual Network Actors and contain information about the Actors cultural proclivities used to translate incoming landscape and network signals into actor reactivity levels representing the overall influence of the incoming signals on the actors interpretation of the signals.
Signal Strength [-1,1] A measure of the strength of the signal at a given edge.
Transmission Efficiency [0,1] The efficiency with with a given edge transfers the signal.
Influence [-1,1] The interpretation of incoming signals by a Network Actor. Influence of a sender Actor on a receiver Actor is determined using a multivariate model incorporating signal strength, the sender and receiver cultural trait vectors, and a set of additional factors TBD.
Network Actor Reactivity [-1,1] Individual influences on a receiver Actor from one or more senders are accumulated into an Actor reactivity using a threshold-based sigmoidal transfer function. This reactivity level in turn is used as the transmission signal strength for any “downstream” connections for which the Actor is the sender. When the Actor in question is a Landscape Actor, this reactivity level is used to guide policy selection in Envision. Positive reactivity levels indicate a positive response to the incoming signals, negative reactivity levels indicate a negative response to the incoming signals.
Network Reorganization A process by which edge connections and/or the factors that control influence are dynamically updated in response to…(TBD)

We can represent a coupled dynamic SES using a social influence network in which a landscape signal is conceptualized as an input signal into the social influence network, which than processes and propogates that signal to generate a collective response. If that collective result influences the strength of the landscape signal, the social system (represented by the network) becomes a coupled system; the nature of the coupling is specified by a functional relationship between the network output signal and the input landscape signal indicating how the SES modifies the landscape via the social influence network.

In a SES analysis, we can vary in network topology, expose the network to a variety of input signal patterns, and explore different feed back couplings to understand coupled system dynamics. Because we are modeling the flow of influence through the network, we make the following assumptions in the network model:

  1. A social influence network is a network of nodes and edges that collectively propagate an input signal through the network to produce an output signal.
  2. Interior Nodes represent individuals and/or institutions in the network.  The reactivity of a node is a measure of the degree to which the individual/institution responds to a landscape signal as processed by the network.
  3. Edges represent the influence flow pathways in the network. Edges are directional, meaning signals flow in a single direction along an edge. The flow of influence is controlled by an edge weight, which measures the strength of the influence of the sender ("from") node on the receiver ("to") node, and the level of reactivity of the sender node. The edge weight is determined dynamically by the influence submodel, described on the "SNIP Model Details" tab above.
  4. Zero to N interior connections are possible for a given node, where N is the maximum node degree.
  5. A node can not be connected to itself.
  6. All nodes have a "from" connection from a single input landscape signal.
Steady State Network Operation
SNIP Model Details
Influence Propagation and Signal Aggregation
Combined Signal Aggregation/Node Reactivity Function
Adding Dynamics

Steady State Network Operation

The social network influence propagation (SNIP) model operationalizes the definitions above into a system that: 1) receives as input one of more landscape signals, 2) processes that signal through a social network consisting of a network actors characterized along cultural trait dimensions, and 3) propagates the landscape signal(s) through the network using the concepts described above. Specifically, the model operates as follows:

  1. A landscape signal (or signals) is generated by Envision or some other source.
  2. The model propagates this signal to all connected Network Actors, which in turn modify and transmit the signal to their “downstream” connections. This process continues until the network “relaxes” into a steady state.
  3. For any Landscape Actors, if their reactivity level from a given landscape signal exceeds a threshold, they make a policy selection based on their level of reactivity and the nature of the specific landscape signal.
  4. If enabled, allow for network reorganization.
  5. Repeat every year of the simulation.

Input signals and node reactivity are in the range [-1,1]. Signal strengths are related to sender node reactivity levels and edge transmission properties, defined in terms of it's ability to propagate influence (see below for details). based on traits of the sender and receiver actor. The following table provides interpretations of the relationship between positive and negative edge weight/"from" node reactivity.

Sender Node Reactivity Level
Input Signal Positive Negative


Positive		 Edge, From Node = Signal
Both the incoming signal and "from" node reactivity are positive, so the resulting signal is positive. This reflects a situation where the influencer ("from" node) is viewed positively by the influencee ("to" node), and the influencer responds positively to the landscape signal. In this case, the effect positively influences the influencee (increases the "to" node reactivity), since the influencee tends to trust the response of the influencer actor, and that actor responded positively to the landscape signal.		 Edge, From Node = Signal
Edge weight is positive, but "from" node reactivity is negative, so the resulting signal is negative. This reflects a situation where the influencer ("from" node) is viewed positively by the influencee ("to" node), and the influencer responds negatively to the landscape signal. In this case, the effect negatively influences the influencee (decreases the "to" node reactivity), since the influencee will tend to trust the influencers' negative response to the landscape signal.


Negative		 Edge, From Node = Signal
Edge weight is negative, but "from" node reactivity is positive, so the resulting signal is negative. This reflects a situation where the influencer ("from" node) is viewed negatively by the influencee ("to" node), and the influencer responds positively to the landscape signal. In this case, the effect negatively influences the influencee (decreases the "to" node reactivity), since the influencee tends to distrust the influencers' positive response to the landscape signal ("if they like it, I don't").		 Edge, From Node = Signal
Both edge weight and "from" node reactivity are negative, so the resulting signal is positive. This reflects a situation where the influencer ("from" node) is viewed negatively by the by the reflect influencee ("to" node), and the influencer responds negatively to the landscape signal. In this case, the signal is positive, because a negative response from an actor who is not trusted can result in a positive response to the landscape signal.

SNIP Model Details

The SNIP Model focuses on how landscape and network signals propagate through a social network. The model considers the following:

  1. Influence propagation between sender and receiver actors along a single connecting edge,
  2. Aggregation of multiple input signals by a single receiver actor into an overall input influence for that actor,
  3. Translation of a receiver's overall input influence on the reactivity level of the receiver actor
  4. Whole-network signal propagation, originating with a landscape signal, based on the reactivity of actors and edge propagation properties

Details of the steps identified above are provided below.

Transmission of Influence Between a Sender Actor and a Receiver Actor

The first step in the model operation involves computing the signal propagation (influence flow) through a given edge connecting a Sender actor and a Receiver actor, for all connections (edges) in the network. For each edge, influence flow is calculated using a simple influence model described below. Three distinct but related transmission efficiency models are available. These are:

  1. Sender-Receiver Submodel: This model assumes transmission efficiency along a given edge is controlled by the relationship between the Sender and Receiver actors. It considers a set of Sender and Receiver actor traits to determine the transmission efficiency of the edge connecting the two actors. For this submodel, the qualities of the signal have no effect on transmission efficiency. The influence flowing along the edge connecting the Sender and Receiver is the product of the transmission efficiency of the edge and the Sender actor's reactivity.


    where:
    • Ts,r is the transmission efficiency between Sender s and Receiver r [0,tmax,1],
    • Is,r is the influence flowing between the Sender and Receiver[-1,1],
    • Rs is the reactivity level of the Sender actor [-1,1],
    • f(Cs,Cr) is a function comparing the cultural trait vectors of the Sender and Receiver. [-1,1] (see exmaple below)

  2. Signal-Receiver Submodel: This model assumes transmission efficiency along a given edge is controlled by the relationship between the signal and a Receiver actor. It considers a set of signal traits and Receiver actor traits to determine the transmission efficency of a Receiver actor's incoming edges. For this submodel, the qualities of the Sender actor have no effect on transmission efficiency or influence flow. The influence flowing along an edge is the product of the transmission efficiency and the signal strength.


    where:
    • Tsig,r is the transmission efficiency of the Receiver's incoming edges [0,tmax,1],
    • Isig,r is the influence flowing along a receiver actor's incoming edge[-1,1],
    • Si is the strength of signal i [-1,1],
    • f(Csig,Cr) is a function comparing the cultural trait vectors of the Signal and Receiver. [-1,1] (see exmaple below)

  3. (Signal+Sender)-Receiver Submodel: This model combines the two submodels above. The transmission efficiency and influence from each of the above submodels is calulated and weighted to produce a flow of influence along the edge between a sender Actor and receiver Actor. For this submodel, the signal qualities, sender traits and receiver traits affect the transmission efficiency and flow of influence along an edge.

    where:
    • ω is weighting factor associated with the Sender-Receiver influence submodel [0,1]

In the case where the "upstream" node is a landscape signal, the Signal-Receiver submodel is always used. In all other cases, any of the three models can be applied to the network.

The transmission efficiency functions f above [0,tmax] describe the relationship between the Sender and Receiver in the case of the Sender-Receiver model, or the relationship between the signal qualities and the Receiver, in the case of the Signal-Receiver model. Efficiencies in the range [0,1) result in signal attenuation, while efficiencies greater than 1.0 result in signal amplification. Transmission efficiencies in both cases are defined using a simple linear-additive model of the form:

where:
  • α, βi are empirically-derived model coefficients
  • Fi is the i-th model factor
  • tmax is the maximum transmission efficiency (assumed to be 1.2 to account for signal amplification)

The factors (F's) in the equation above are to be determined, and represent factors that influence signal transmission; the model coefficients represents the relative weighting of those factors.

An Example

Assume that two factors affect how much influence a sender actor or landscape signal has on a receiver actor - 1) Similarity of the Sender and Receiver actors (Sender-Receiver Model) or Signal and Receiver actor (Signal-Receiver Model) in cultural dimension space (as expressed in their cultural trait vectors), and 2) availability of resources (e.g. money) to support efforts at addressing the landscape signal. The transmission efficiency of the edge would be computed using these two factors as input; therefore, we would need to be able to quantify each factor for this edge.

We could quantify similarity using the Euclidean distance between the Sender/Signal and Receiver actor’s cultural trait vectors Cs and Cr, scaled to the maximum possible distance in the N-dimensional cultural trait space (e.g. the highest possible dissimilarity between the two actors) as follows:

where:
  • Similarity(s,r) is the similarity measure between the sender actor (s) and the receiver actor (r) [0-1],
  • Cs is the Sender actor's cultural trait vector (Sender-Receiver model) or the signals' cultural trait vector (Signal-Receiver model)
  • r are the Receiver actor’s cultural trait vector [-1,1],
  • Dmax is the maximum possible distance between two vectors in the N-dimensional cultural trait space:

Available resources could be quantified as a budget the sender actor can bring to the receiver actor (we assume a constant $ amount R(s,r)):

The overall influence transmission efficiency model in this case is:

where:
  • R(s,r) is a measure of the resources the sender actor (r) can provide the receiver actor (s).

Aggregation of Multiple Influences to an Overall Signal Strength for a Receiver Actor

A receiver actor may receive 0 or more incoming sender signals. These are combined into an overall signal reflecting the strength and number of the individual influence signals being received, in combination with an Actor influence sensitivity parameter (σ). The resulting relationship is shown on the chart to the right and is defined below:


where:
  • Irtotal is the sum of all inputs into the receiver actor [-∞,∞]
  • Srtotal is the modified input signal, reflecting diminishing returns from multiple signals [-1,1]
  • σ is the actor’s sensitivity to multiple influence signals [1, ∞]

In the chart to the right, the horizontal axis is the sum of the individual influences being received by the actor, and the vertical axis is the resulting signal strength (Srtotal) perceived by the actor.

This signal strength in turn is used to determine an reactivity level for the receiver actor, using the Actor (Node) Reactivity Function given in the next section below.

Actor Sensitivity to Multiple Inputs (σ) 75

Maximum Transmission Efficiency (tmax) 75

Actor (Node) Reactivity Function

Once the receiver actor processes it's incoming influence(s) into a combined influence signal, the signal is translated to an actor (node) reactivity level (in the range [-1,1]), using a logistic transfer function with a parameter reflecting actor reactivity sensitivity (b) and an actor reactivity threshold (τ) shown in the chart to the right and described below. Any input signal whose signal strength is below the reactivity threshold is unable to reactivity the node (i.e. node reactivity level = 0)


where:
  • b is the Actor Reactivity Sensitivity, and
  • τ is the reactivity threshold

The sigmoidal nature of this function results in a response that is relatively low (or zero) when the combined input influence signal is relatively weak - it takes a reasonably strong signal to get the actor over the "motivation" hump - but at higher incoming signal strength, the actor response increases fairly quickly.

Actor (Node) Reactivity Sensitivity (b) 4

Reactivity Threshold (τ) 4

Combined Signal Aggregation/Node Reactivity Function

Combines the functions above on a single chart

Maximum Transmission Efficiency

Actor Influence Sensitivity (σ) 75

Maximum Transmission Efficiency (tmax) 1.2

Actor (Node) Reactivity Sensitivity (b) 4

Reactivity Threshold (τ) 4

Adding Dynamic Network Behaviors

The equations above represent steady state (equilibrium) solutions to the network activation functions. Adding dynamics to this model requires the inclusion of additional time-variant terms that represent the dynamic aspect(s) of real influence networks. Dynamic aspects of influence network function of potential interest include:

  • Time lags in signal propagation through the network;
  • Signal degradation as it moves through the network;
  • Reactivity Fatigue (decreasing sensitivity to incoming signals)

We consider these as follows:

Time Lags in Signal Propagation

In a dynamic simulation, the SNIP model doesn't explicitly capture the passage of time; rather, it proceeds in a series of steps (termed cycles) that reflect the movement of the signal across an edge from a sender to a receiver. Edges can optionally define a signal transit time as an edge attribute that indicates how many cycles it takes for a signal to traverse the edge. These transit times are defined in the network definition file.

Signal Degradation during Transmission

Signal degradation during transmission is based on the concept of a physical signal moving from a starting location to various nodes throughout the network based on edge topology. In influence transmission networks, the signal can be thought of as starting at a specified point in the network, and then moving to the immediate neighborhood around that source, and then propagating throughout the network, based on edge topology, from neighbor to neighbor.

To represent this concept in the SNIP model, we assume that signal propagation starts at the landscape node(s), and initally proceeds through each landscape edge to the immediate neighbors of the landscape node(s). Those neighbors, in the next cycle, in turn transmit the signal to their immediate neighborhood, a pattern which repeats through a series of cycles. To model the signal degredation during each "jump" from one actor to the next, we assume the input signal received by an actor is diminished as a linear function of the number of edges/cycles the signal has traversed since the initial signal was generated. Mathematically, we focus on modifying the individual influence signals being tranmitted along a given edge between a given sender and receiver, Is,r in the equations above. We will subject this to an inverse linear decay process that scales with the number of cycles taken so far, quantified as follows:

where:
  • I*s,r is the degraded signal,
  • Is,r is the undegraded signal determined from the transmission efficiency and sender reactivity,
  • kd is fraction of the signal that is degraded at each cycle.

Actor Reactivity Fatigue

Reactivity fatigue means that as actor is exposed to incoming influence signals, over time their sensitivity to those signals goes down. There are two model parameters described above that are relevant to this behavior: 1) b, the Actor reactivity sensitivity, and τ, the reactivity threshold. The SNIP model allows for varying an actor's reactivity sensitivity based on the given actor's average reactivity in the last 10 cycles. This is represented in the model by an inverse linear relationship of the form:

where:
  • b* is the "fatigued" Actor reactivity sensitivity for Actor i,
  • b is the "unfatigued" Actor reactivity sensitivity for Actor i, and
  • fb is "fatigue" scalar - higher values indicate a stronger fatigue response, lower values a less fatigu0e response, and zero indicating no fatigue response,
  • Rma10 is the 10-cycle moving average of Actor i's reactivity.

SNIP Model Operation

The SNIP model operates in the following steps:
  1. Initialization: When a network model is first loaded, the following steps are executed:
    1. A network is created using the settings, node definitions, and edge definitions in the network definition file.
    2. The 'input" node is identified.
    3. The network is "reset", which
      1. Sets all Actor nodes "state" to active.
      2. Sets all Actor nodes "reactivity", "influence" and other node data to 0.
      3. Sets all edges "state" to active.
      4. Sets all edges data ("activeCycles", "transEff", "influence", "signalStrength") to 0.
      5. Sets all edges "signalTraits" to the trait values of input signal.
      6. Sets the input signal node's "reactivity" to 1.0.
      7. Set all input signal edges' "signalStrength" to 1.0.
    4. The network is solved for the equilibrium state (see (2) below).
    5. Initial network stats are computed and displayed.
  2. Solving for the Network Equilibrium State: This step involves calculating reactivities, influences, and related node and edge states for the current network input conditions. This is an iterative process that "relaxes" the network state until it reaches an equilibrium state, meaning successive iterations do not change any node/edge states. The steps in this process include:
    1. For all ACTIVE EDGES, compute edge transmission efficiencies using the currently selected Influence Submodel. Note that these do not change during a given cycle.
    2. Iteratively perform the following steps until the network state converges:
      1. For all ACTIVE EDGES, compute influence associated with the edge
      2. For all ACTIVE ACTOR NODES, compute the reactivity of the node by:
        1. Summing the influences coming "into" the node and aggregating those inputs into an aggregated input signal.
        2. Processing the aggregated input signal into a reactivity level using an activation function.
        3. Updating the node's "reactivity" state.
    3. Once the network has converged, calculate the influence associated with each node by summing the influences associated with each outgoing edge from the node.
  3. Running a Simulation: Running a simulation involves propating a landscape signal through the network topology, starting with the landscape signal (input) node, to its neighbors, and then their neighbors, etc. through a set number of cycles. The steps involved in running a simulation are:
    1. Initializing the simulation. This step performs the following tasks:
      1. Clearing out an prior network/watch list reports.
      2. Resetting the network (see Item 1.3 above).
      3. Set all ACTOR NODES to an inactive state.
      4. Set all EDGES to an inactive state.
      5. Set all EDGES "signalStrength" to 0, and their "signalTraits" to null.
    2. Once initialization is complete, the simulation starts running through cycles. In each cycle, the following steps are employed:
      1. The input signal level is established, for all input signals, based on the network configuration setup, and the input node's "reactivy" is set to this input level.
      2. The signal is propagated through the network for one cycle as follows:
        1. For each ACTIVE NODE, outgoing edges that are currently inactive and convert them to an "activating" state. This signals that the edge will become active once the number of cycles indicated by its "transitTime" attribute have passed.
        2. The network is then solved for its equilibrium state, reflecting any new edges that have become activated since the last cycle.
        3. Once the new equilibrium state is established, each edge that is either "active" or "activating" has its "activeCycles" attribute incremented by one.
        4. For each "activating" edge, if the number of active cycles exceed its "transitTime", it is set to an "active" state. Upon becoming active, the edge's "signalStrength" attribute is set to the "reactivity" of its source (upstream) node, and its "signalTraits" attribute set to the input signals "traits". Additionally, the edge's target (downstream) node is set to an "active" state.
        5. For every ACTIVE NODE, the reactivity history moving window is updated
      3. At the end of the cycles,network statistics are collected and the user interface updated.
      4. This process is continued until all cycles in the simulation have been run.

Network Learning and Self-Organization

Optionally, networks can modify their topologies in response to being activated.  This is accomplished through application of a learning rule that describes how edge weights change in response to connecting node activation. An example of a simple learning rule is "if two connected nodes are simultaneously activated, increase the strength (edge weight) of the connection between them; otherwise, decrease it."

The chart to the right shows an example of a learning rule that captures the description above in mathematical form. It maps the difference in reactivity levels between two connected nodes (x axis) to the corresponding edge weight adjustment that is applied as part of the learning algorithm. This edge eadjustment is provided for all network edges. Note that nodes that respond to an external stimulus (landscape signal in the case of a SES) similarly (have similar node reactivity levels) will have low Node Deltas, and map to the left side of the chart; dissimilar node reactivity result in a reduction in the connecting edge weight, and map to the right hand side of the chart.

Tuning Parameter 4