Tion steps (for which we could speak of an energy entrapment principle). Notice how the term m ?in (6) makes the change in the connection wi;j ?proportional to j;p the quantity of free energy by node m ?in favor of node m ?. The whole learning process, i;p i;p which essentially consists of a progressive adjustment of the connections aimed at the global minimization of energy, may be seen as a complex juxtaposition of Pinometostat supplier phases of get PX-478 acceleration and deceleration of velocities of the learning signals (adaptations Dwi;j ?and Dvi ?) inside the ANN connection matrix. To get a clearer understanding of this feature of the AutoCM learning mechanics, begin by considering its convergence condition:n!lim vi ??C??Indeed, when vi ??C, then Dvi ??0 (according to Eq 2), and m ??0j8p 2 M (accordj;p ing to Eq 1) and, subsequently, Dwi;j ??0 (as from Eq 6): the AutoCM then converges. The matrix w (Eq 7), then, represents the AutoCM knowledge about the whole dataset. Now, if we consider C as a limit value for all the weights of the w matrix, we can write: wi;j ?wj;i ; 2 0 0 wj;i ?wi;j ; wi;j ???if i 6?j;di;j ?C ?wi;j di;i ?0:The new matrix d is a squared symmetric matrix, where the main diagonal entries are null (i.e., they represent the zero distance of each variable from itself), and where the off-diagonal entries represent `distances’ between each couple of variables.AutoCM and Minimum Spanning Tree (MST)Eq (9) transforms the squared weight matrix of AutoCM into a squared matrix of distances among nodes [25]. Each distance between a pair of nodes may therefore be regarded as the weighted edge between these pairs of nodes in a suitable graph-theoretical representation, so that the matrix d itself may be analyzed through the graph theory toolbox. A graph is a mathematical abstraction that is useful for solving many kinds of problems. Fundamentally, a graph consists of a set of vertices, and a set of edges, where an edge is an object that connects two vertices in the graph. More precisely, a graph is a pair (V, E), where V is a finite set and E is a binary relation on V, to which it is possible to associate scalar values (in this case, the distances di,j). V is called a vertex set which elements are called vertices. E is a collection of edges, where an edge is a pair (u, v) with u, v belonging to V. In a directed graph, edges are ordered pairs, connecting a source vertex to a target vertex. In an undirected graph, edges are un-ordered pairs and connect the two vertices in both directions, hence in an undirected graph (u,v) and (v, u) are two ways of writing the same edge. The graph-theoretical representation is not constrained by any a priori semantic restriction: it does not say what a vertex or edge actually represents. They could be cities with connecting roads, or web-pages with hyperlinks, and so on. These semantic details are irrelevant toPLOS ONE | DOI:10.1371/journal.pone.0126020 July 9,7 /Data Mining of Determinants of IUGRdetermine the graph structure and properties; the only thing that matters is that a specific graph may be taken as a proper representation of the phenomenon under study, to justify attention on that particular mathematical object. An adjacency-matrix representation of a graph is a 2-dimensional VxV array, where rows represent the list of vertices and columns represent edges among vertices. To each element in the array a Boolean value, describing whether the edge (u,v) is in the graph, is assigned. A distance matrix among V vert.Tion steps (for which we could speak of an energy entrapment principle). Notice how the term m ?in (6) makes the change in the connection wi;j ?proportional to j;p the quantity of free energy by node m ?in favor of node m ?. The whole learning process, i;p i;p which essentially consists of a progressive adjustment of the connections aimed at the global minimization of energy, may be seen as a complex juxtaposition of phases of acceleration and deceleration of velocities of the learning signals (adaptations Dwi;j ?and Dvi ?) inside the ANN connection matrix. To get a clearer understanding of this feature of the AutoCM learning mechanics, begin by considering its convergence condition:n!lim vi ??C??Indeed, when vi ??C, then Dvi ??0 (according to Eq 2), and m ??0j8p 2 M (accordj;p ing to Eq 1) and, subsequently, Dwi;j ??0 (as from Eq 6): the AutoCM then converges. The matrix w (Eq 7), then, represents the AutoCM knowledge about the whole dataset. Now, if we consider C as a limit value for all the weights of the w matrix, we can write: wi;j ?wj;i ; 2 0 0 wj;i ?wi;j ; wi;j ???if i 6?j;di;j ?C ?wi;j di;i ?0:The new matrix d is a squared symmetric matrix, where the main diagonal entries are null (i.e., they represent the zero distance of each variable from itself), and where the off-diagonal entries represent `distances’ between each couple of variables.AutoCM and Minimum Spanning Tree (MST)Eq (9) transforms the squared weight matrix of AutoCM into a squared matrix of distances among nodes [25]. Each distance between a pair of nodes may therefore be regarded as the weighted edge between these pairs of nodes in a suitable graph-theoretical representation, so that the matrix d itself may be analyzed through the graph theory toolbox. A graph is a mathematical abstraction that is useful for solving many kinds of problems. Fundamentally, a graph consists of a set of vertices, and a set of edges, where an edge is an object that connects two vertices in the graph. More precisely, a graph is a pair (V, E), where V is a finite set and E is a binary relation on V, to which it is possible to associate scalar values (in this case, the distances di,j). V is called a vertex set which elements are called vertices. E is a collection of edges, where an edge is a pair (u, v) with u, v belonging to V. In a directed graph, edges are ordered pairs, connecting a source vertex to a target vertex. In an undirected graph, edges are un-ordered pairs and connect the two vertices in both directions, hence in an undirected graph (u,v) and (v, u) are two ways of writing the same edge. The graph-theoretical representation is not constrained by any a priori semantic restriction: it does not say what a vertex or edge actually represents. They could be cities with connecting roads, or web-pages with hyperlinks, and so on. These semantic details are irrelevant toPLOS ONE | DOI:10.1371/journal.pone.0126020 July 9,7 /Data Mining of Determinants of IUGRdetermine the graph structure and properties; the only thing that matters is that a specific graph may be taken as a proper representation of the phenomenon under study, to justify attention on that particular mathematical object. An adjacency-matrix representation of a graph is a 2-dimensional VxV array, where rows represent the list of vertices and columns represent edges among vertices. To each element in the array a Boolean value, describing whether the edge (u,v) is in the graph, is assigned. A distance matrix among V vert.