Emergence of Leadership in Communication

Abstract

We study a neuro-inspired model that mimics a discussion (or information dissemination) process in a network of agents. During their interaction, agents redistribute activity and network weights, resulting in emergence of leader(s). The model is able to reproduce the basic scenarios of leadership known in nature and society: laissez-faire (irregular activity, weak leadership, sizable inter-follower interaction, autonomous sub-leaders); participative or democratic (strong leadership, but with feedback from followers); and autocratic (no feedback, one-way influence). Several pertinent aspects of these scenarios are found as well-e.g., hidden leadership (a hidden clique of agents driving the official autocratic leader), and successive leadership (two leaders influence followers by turns). We study how these scenarios emerge from inter-agent dynamics and how they depend on behavior rules of agents-in particular, on their inertia against state changes.

Fig 1. Laissez-faire leadership.

In Eq (6) we set f = 1 and α = 1. The behavioral noise is weak: η = 0.07; cf. Eqs (18) and (19). The intergent coupling q = 2 is sizable, but is smaller than Eq (17). The observation time: T = 600; see Eq (20). Black points (upper curve): the average weights ${\overline{\tau }}_{kL}$ by which the leader L influences other agents; see Eq (20). Red points (lower curve): ${\overline{\tau }}_{kS}$ that quantify the influence of the leading sub-leader S on other agents. Here ${\overline{\tau }}_{kS}$ and ${\overline{\tau }}_{kL}$ were separately arranged in the decreasing order over k.

Fig 2. Laissez-faire leadership.

The same parameters as in Fig 1, but for the weights ${\overline{\tau }}_{Lk}$ and ${\overline{\tau }}_{Sk}$ that quantify the influence of followers on the leader L, and on the sub-leader S, respectively. Again, ${\overline{\tau }}_{Sk}$ and ${\overline{\tau }}_{Lk}$ were separately arranged in decreasing order. Note that the influence of L and on S, and S on L are comparable: ${\overline{\tau }}_{SL}=0.545378$ and ${\overline{\tau }}_{LS}=0.535170$.

Fig 3. Laissez-faire leadership.

The same parameters as in Fig 1. Collective activity $m\left(t\right)={\sum }_{i=1}^N{m}_i\left(t\right)$ versus time t. It is seen that m(t) displays irregular (noisy) behavior.

Fig 4. Laissez-faire leadership.

The same parameters as in Figs 1 and 3 Collective activity $m\left(t\right)={\sum }_{i=1}^N{m}_i\left(t\right)$ versus time t, but for a weaker noise with magnitude η = 0.01.

Fig 5. Participative leadership.

Emergent network structures according to Eqs Eqs (1)–(11) and (13). Parameters are chosen from Eqs (16) and (17) and $q>{\mathcal{Q}}^{-}$. Single Participative leader: α = β = 1; see Eqs (6) and (7). The leader L (red square) has the highest score (≃500) and stimulates all other agents (followers, green squares) with the maximal weight τ_iL = 1 (bold arrows). Followers (green squares) have different credibilities ${\sigma }_i=\mathcal{O}(1/N)$; each of them stimulates the leader with weights ${\tau }_{Li}=\mathcal{O}(1/N)$: a follower with a larger score influences the reader stronger.

Fig 6. Participative leadership: single emergent leader.

Distribution of stationary credibilities σ_k (blue points, upper curve) and weights τ_1k (red points, lower curve) by which the agent with rank k (N ≥ k ≥ 2) influences the leader (k = 1). The agents are ranked according to their final score: k = 1 is the highest-score agent (leader), k = N is the lowest score agent. Eqs (1)–(9) and (13) are solved for Eq (16) and q = 2.5. The dynamics was followed by 200 time-units.

Fig 7. Participative leadership.

Single emergent leader. The collective activity $m\left(t\right)=\frac{1}{N}{\sum }_{k=1}^N{m}_k\left(t\right)$ as a function of time t for q = 2.7 (other parameters are the same as in Fig 6). Black points (two straight lines): the noiseless situation η = 0. Magenta points: η = 0.05; cf. Eqs (18) and (19). In the noiseless situation m(t) takes only two values 0.01 (the leader is active) and 0.99 (followers are active). For the noisy situation m(t) assumes two well-separated sets of values at ∼0.1 and ∼0.9, respectively.

Fig 8. Participative leadership.

Hierarchic leadership: α = 2, β = 1. (Other parameters are the same as in Fig 5.) The followers are divided into two groups, strongly driven by respectively leader (red) and sub-leader (blue). The feedback is collected by the leader only.

Fig 9. Autocratic leadership.

Single autocratic leader: α = 1, β = 0 and $q>{\mathcal{Q}}^{-}$; see Eqs (6), (7), (14) and (15). Other parameters are chosen according to Eq (16). The leader (red square) stimulates others (green squares) and is stimulated by the helper (blue square). All these stimulations have the maximal weight equal to 1. All other agents are passive spectators with zero score.

Fig 10. Autocratic leadership.

Hidden leadership: α = 1.5, β = 0 (other parameters are those of Fig 9). The highest-score agent (red square) is driven by a circle of agents that drive each other cyclically (grey squares). The official leader (red square) has the largest number of followers (green squares), though each gray agent can have its own followers. All followers are driven by the maximal weight, have neglegible credibilities and do not feedback.

Fig 11. Coalition of two leaders.

Here α = 0.25, β = 1 and Eq (16). The red (blue) agent has the highest (one but highest) score. However, the real leader is now the blue agent, since it stimulates all other agents with the weight equal to 1. The red agent stimulates the blue one with the weight 1, and all other agents with the weight ≃1/N. All green agents are passive spectators with score close to zero.